Properties

Degree $2$
Conductor $10830$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s − 4·7-s + 8-s + 9-s − 10-s + 6·11-s − 12-s + 4·13-s − 4·14-s + 15-s + 16-s + 6·17-s + 18-s − 20-s + 4·21-s + 6·22-s + 6·23-s − 24-s + 25-s + 4·26-s − 27-s − 4·28-s − 2·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s − 0.288·12-s + 1.10·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.223·20-s + 0.872·21-s + 1.27·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.755·28-s − 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10830\)    =    \(2 \cdot 3 \cdot 5 \cdot 19^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{10830} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10830,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.779700665\)
\(L(\frac12)\) \(\approx\) \(2.779700665\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 + T \)
19 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.31286803719075, −16.17744689422631, −15.39663464320273, −14.73365869380280, −14.38671510425489, −13.39814226296921, −13.14873990697172, −12.46656726380677, −11.95762189767889, −11.46271383804755, −10.95996656470077, −10.05537902870901, −9.623977513229262, −8.956025760850523, −8.181975092094240, −7.222402213725653, −6.669220319867293, −6.315032180732699, −5.685267879764736, −4.876802130821405, −3.888088212230882, −3.574254697840382, −3.002914704114474, −1.502819293810445, −0.7843023436907310, 0.7843023436907310, 1.502819293810445, 3.002914704114474, 3.574254697840382, 3.888088212230882, 4.876802130821405, 5.685267879764736, 6.315032180732699, 6.669220319867293, 7.222402213725653, 8.181975092094240, 8.956025760850523, 9.623977513229262, 10.05537902870901, 10.95996656470077, 11.46271383804755, 11.95762189767889, 12.46656726380677, 13.14873990697172, 13.39814226296921, 14.38671510425489, 14.73365869380280, 15.39663464320273, 16.17744689422631, 16.31286803719075

Graph of the $Z$-function along the critical line